# Stochastic Volatility and GARCH: Do Squared End-of-Day Returns Provide Similar Information?

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## Abstract

**:**

## 1. Introduction

## 2. Previous Work and Econometric Models

#### 2.1. Stochastic Volatility

#### 2.2. ARCH and GARCH

#### 2.3. Realised Volatility

#### 2.4. Historical Volatility Model

#### 2.5. Heterogenous Autoregressive Model (HAR)

## 3. Results of the Analysis

#### 3.1. Preliminary Analysis

#### 3.2. SV and GARCH Estimates

#### 3.3. Mincer–Zarnowitz Tests

#### 3.4. Further Analysis

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Mincer–Zarnowitz rolling regression slope coefficients bounded by two standard deviations.

Mean | Median | Minimum | Maximum | Standard Deviation | Skewness | Excess Kurtosis | |
---|---|---|---|---|---|---|---|

SPRET | 0.000136 | 0.000540 | −0.126700 | 0.106420 | 0.012485 | −0.360540 | 10.874 |

SPRV5 | 0.000112 | 0.000047 | 0.000001 | 0.007748 | 0.000269 | 10.6520 | 188.60 |

DJRET | 0.000150 | 0.000488 | −0.138070 | 0.107540 | 0.011946 | −0.376640 | 13.380 |

DJRV5 | 0.000114 | 0.000049 | 0.000001 | 0.008624 | 0.000287 | 12.0870 | 238.02 |

STOXX50RET | −0.000098 | 0.000236 | −0.120050 | 0.105540 | 0.014399 | −0.225470 | 5.8411 |

STOXX50RV5 | 0.000161 | 0.000081 | 0.000000 | 0.010827 | 0.000336 | 12.0260 | 256.24 |

KEY: | |||||||

SPRET | Continuously compounded close to close return on the S&P500 Index | ||||||

SPRV5 | Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man | ||||||

DJRET | Continuously compounded close to close return on the DOWJONES Index | ||||||

DJRV5 | Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man | ||||||

STOXX50RET | Continuously compounded close to close return on the S&P500 Index | ||||||

STOXX50RV5 | Daily realised volatility on the S&P500 Index sampled at 5 min intervals provided by Oxford Man |

Summary of 1000 MCMC Draws after Burn in of 1000 | |||||
---|---|---|---|---|---|

Prior Distributions | |||||

$\mu \sim normal$ | mean = 0 | S.D. = 100 | |||

$(\varphi +1)/2\sim \beta $ | ${a}_{0}=5$ | ${b}_{0}=1.5$ | |||

${\sigma}^{2}\sim {\chi}^{2}(df=1)$ | |||||

S&P500 | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −9.4565 | 0.15973 | −9.7172 | −9.4562 | −9.193 |

$\varphi $ | 0.9803 | 0.00377 | 0.9739 | 0.9804 | 0.986 |

$\sigma $ | 0.2154 | 0.01526 | 0.1902 | 0.2152 | 0.241 |

$exp(\mu /2)$ | 0.0089 | 0.00071 | 0.0078 | 0.0088 | 0.010 |

${\sigma}^{2}$ | 0.0466 | 0.00659 | 0.0362 | 0.0463 | 0.058 |

DOWJONES | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −9.5491 | 0.14899 | −9.7890 | −9.7890 | −9.3053 |

$\varphi $ | 0.9784 | 0.00393 | 0.9718 | 0.9785 | 0.9846 |

$\sigma $ | 0.2222 | 0.01509 | 0.1982 | 0.2214 | 0.2470 |

$exp(\mu /2)$ | 0.0085 | 0.00063 | 0.0075 | 0.0084 | 0.0095 |

${\sigma}^{2}$ | 0.0496 | 0.00676 | 0.0393 | 0.0490 | 0.0610 |

STOXX50 | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −8.969 | 0.15544 | −9.2223 | −8.969 | −8.719 |

$\varphi $ | 0.983 | 0.00346 | 0.9774 | 0.983 | −8.719 |

$\sigma $ | 0.177 | 0.01362 | 0.1556 | 0.176 | 0.200 |

$exp(\mu /2)$ | 0.011 | 0.00088 | 0.0099 | 0.011 | 0.200 |

${\sigma}^{2}$ | 0.031 | 0.00484 | 0.0242 | 0.031 | 0.040 |

Coefficients | Standard Error | T Statistic | |
---|---|---|---|

S&P500 | |||

$\mu $ | 0.00071466 | 0.0001000 | 7.144 *** |

$\omega $ | 0.0000013037 | 0.0000002911 | 4.479 *** |

${\alpha}_{1}$ | 0.12429 | 0.0118 | 10.530 *** |

${\beta}_{1}$ | 0.87470 | 0.01081 | 80.918 *** |

DOWJONES | |||

$\mu $ | 0.00028945 | 0.0001063 | 2.723 *** |

$\omega $ | 0.000001861 | 0.000000258 | 7.212 *** |

${\alpha}_{1}$ | 0.12121 | 0.009328 | 12.995 *** |

${\beta}_{1}$ | 0.86583 | 000.9399 | 92.123 *** |

STOXX50 | |||

$\mu $ | 0.000093821 | 0.0001411 | 0.665 |

$\omega $ | 0.0000024108 | 0.000000407 | 5.911 *** |

${\alpha}_{1}$ | 0.099385 | 0.008582 | 11.580 *** |

${\beta}_{1}$ | 0.89020 | 0.009059 | 98.266 *** |

S&P500 | ||||
---|---|---|---|---|

OLS, using observations 2000-01-06–2020-04-30 (T = 5098) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −0.000193903 | 4.80241 × 10${}^{-6}$ | −40.38 | 0.0000 |

STVOL_1 | 0.000305766 | 4.04560 × 10${}^{-6}$ | 75.58 | 0.0000 |

Mean dependent var | 0.000112 | S.D. dependent var | 0.000269 | |

Sum squared resid | 0.000174 | S.E. of regression | 0.000185 | |

${R}^{2}$ | 0.528511 | Adjusted ${R}^{2}$ | 0.528418 | |

$F(1,5096)$ | 5712.306 | P-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.355673 | Durbin–Watson | 1.288578 | |

OLS, using observations 2000-01-06–2020-04-30 (T = 5098) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −0.000186320 | 5.58228 × 10${}^{-6}$ | −33.38 | 0.0000 |

garchh_t_1 | 0.000282384 | 4.54881 × 10${}^{-6}$ | 62.08 | 0.0000 |

Mean dependent var | 0.000112 | S.D. dependent var | 0.000269 | |

Sum squared resid | 0.000210 | S.E. of regression | 0.000203 | |

${R}^{2}$ | 0.430599 | Adjusted ${R}^{2}$ | 0.430488 | |

$F(1,5096)$ | 3853.762 | P-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.432641 | Durbin–Watson | 1.134608 | |

OLS, using observations 2000-02-02–2020-04-30 (T = 5079) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 3.02687 × 10${}^{-5}$ | 2.84918 × 10${}^{-6}$ | 10.62 | 0.0000 |

sq_DMSPRET_1 | 0.174138 | 0.00556030 | 31.32 | 0.0000 |

sq_DMSPRET_2 | 0.0933966 | 0.00559548 | 16.69 | 0.0000 |

sq_DMSPRET_3 | 0.0527851 | 0.00588575 | 8.968 | 0.0000 |

sq_DMSPRET_4 | 0.0280498 | 0.00588689 | 4.765 | 0.0000 |

sq_DMSPRET_5 | 0.0591621 | 0.00588274 | 10.06 | 0.0000 |

sq_DMSPRET_6 | 0.0387770 | 0.00590617 | 6.565 | 0.0000 |

sq_DMSPRET_7 | 0.0127639 | 0.00593373 | 2.151 | 0.0315 |

sq_DMSPRET_8 | 0.0128494 | 0.00591714 | 2.172 | 0.0299 |

sq_DMSPRET_9 | 0.0460289 | 0.00591518 | 7.781 | 0.0000 |

sq_DMSPRET_10 | 0.0108100 | 0.00588990 | 1.835 | 0.0665 |

sq_DMSPRET_11 | −0.0125045 | 0.00589057 | −2.123 | 0.0338 |

sq_DMSPRET_12 | 0.00683578 | 0.00591643 | 1.155 | 0.2480 |

sq_DMSPRET_13 | 0.000837936 | 0.00591795 | 0.1416 | 0.8874 |

sq_DMSPRET_14 | −0.00804767 | 0.00593584 | −1.356 | 0.1752 |

sq_DMSPRET_15 | −0.00239430 | 0.00590724 | −0.4053 | 0.6853 |

sq_DMSPRET_16 | −0.0108916 | 0.00588602 | −1.850 | 0.0643 |

sq_DMSPRET_17 | 0.00482912 | 0.00588926 | 0.8200 | 0.4123 |

sq_DMSPRET_18 | 0.0125097 | 0.00593287 | 2.109 | 0.0350 |

sq_DMSPRET_19 | 0.0123520 | 0.00564065 | 2.190 | 0.0286 |

sq_DMSPRET_20 | −0.00717624 | 0.00560040 | −1.281 | 0.2001 |

Mean dependent var | 0.000112 | S.D. dependent var | 0.000269 | |

Sum squared resid | 0.000169 | S.E. of regression | 0.000183 | |

${R}^{2}$ | 0.541977 | Adjusted ${R}^{2}$ | 0.540166 | |

$F(20,5058)$ | 299.2563 | P-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.283092 | Durbin–Watson | 1.433782 | |

OLS, using observations 2000-01-06–2020-04-30 (T = 5094) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −0.000198625 | 5.37151 × 10${}^{-6}$ | −36.98 | 0.0000 |

SVDJ_1 | 0.000328020 | 4.75466 × 10${}^{-6}$ | 68.99 | 0.0000 |

Mean dependent var | 0.000114 | S.D. dependent var | 0.000287 | |

Sum squared resid | 0.000216 | S.E. of regression | 0.000206 | |

${R}^{2}$ | 0.483125 | Adjusted ${R}^{2}$ | 0.483024 | |

$F(1,5092)$ | 4759.515 | P-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.331936 | Durbin–Watson | 1.336075 | |

OLS, using observations 2000-01-06–2020-04-30 (T = 5094) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −0.000170947 | 6.16017 × 10${}^{-6}$ | −27.75 | 0.0000 |

Djht_1 | 0.000280420 | 5.18265 × 10${}^{-6}$ | 54.11 | 0.0000 |

Mean dependent var | 0.000114 | S.D. dependent var | 0.000287 | |

Sum squared resid | 0.000266 | S.E. of regression | 0.000228 | |

${R}^{2}$ | 0.365056 | Adjusted ${R}^{2}$ | 0.364932 | |

$F(1,5092)$ | 2927.610 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.434841 | Durbin–Watson | 1.130290 | |

OLS, using observations 2000-01-04–2020-04-30 (T = 5075) | ||||

Dependent variable: lrv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.0405870 | 0.00329492 | 12.32 | 0.0000 |

SQDMDJRET_1 | 0.000165523 | 6.55973 × 10${}^{-6}$ | 25.23 | 0.0000 |

SQDMDJRET_2 | 0.000101750 | 6.61717 × 10${}^{-6}$ | 15.38 | 0.0000 |

SQDMDJRET_3 | 5.22487 × 10${}^{-5}$ | 6.99793 × 10${}^{-6}$ | 7.466 | 0.0000 |

SQDMDJRET_4 | 2.10587 × 10${}^{-5}$ | 6.99864 × 10${}^{-6}$ | 3.009 | 0.0026 |

SQDMDJRET_5 | 6.58804 × 10${}^{-5}$ | 7.00022 × 10${}^{-6}$ | 9.411 | 0.0000 |

SQDMDJRET_6 | 4.64120 × 10${}^{-5}$ | 7.00145 × 10${}^{-6}$ | 6.629 | 0.0000 |

SQDMDJRET_7 | 1.56715 × 10${}^{-6}$ | 7.13269 × 10${}^{-6}$ | 0.2197 | 0.8261 |

SQDMDJRET_8 | 1.54284 × 10${}^{-6}$ | 7.11737 × 10${}^{-6}$ | 0.2168 | 0.8284 |

SQDMDJRET_9 | 4.60489 × 10${}^{-5}$ | 7.11644 × 10${}^{-6}$ | 6.471 | 0.0000 |

SQDMDJRET_10 | −2.98185 × 10${}^{-6}$ | 7.06670 × 10${}^{-6}$ | −0.4220 | 0.6731 |

SQDMDJRET_11 | −1.35137 × 10${}^{-5}$ | 7.06710 × 10${}^{-6}$ | −1.912 | 0.0559 |

SQDMDJRET_12 | −2.77575 × 10${}^{-6}$ | 7.11818 × 10${}^{-6}$ | −0.3900 | 0.6966 |

SQDMDJRET_13 | 5.94860 × 10${}^{-6}$ | 7.11930 × 10${}^{-6}$ | 0.8356 | 0.4034 |

SQDMDJRET_14 | −8.20421 × 10${}^{-6}$ | 7.13518 × 10${}^{-6}$ | −1.150 | 0.2503 |

SQDMDJRET_15 | −8.26171 × 10${}^{-6}$ | 7.00344 × 10${}^{-6}$ | −1.180 | 0.2382 |

SQDMDJRET_16 | −4.30980 × 10${}^{-6}$ | 7.00422 × 10${}^{-6}$ | −0.6153 | 0.5384 |

SQDMDJRET_17 | 6.42163 × 10${}^{-6}$ | 7.00281 × 10${}^{-6}$ | 0.9170 | 0.3592 |

SQDMDJRET_18 | 2.34879 × 10${}^{-5}$ | 7.08619 × 10${}^{-6}$ | 3.315 | 0.0009 |

SQDMDJRET_19 | 2.02923 × 10${}^{-5}$ | 6.69703 × 10${}^{-6}$ | 3.030 | 0.0025 |

SQDMDJRET_20 | −4.02896 × 10${}^{-6}$ | 6.63951 × 10${}^{-6}$ | −0.6068 | 0.5440 |

Mean dependent var | 0.113763 | S.D. dependent var | 0.287164 | |

Sum squared resid | 230.9808 | S.E. of regression | 0.213782 | |

${R}^{2}$ | 0.447966 | Adjusted ${R}^{2}$ | 0.445781 | |

$F(20,5054)$ | 205.0615 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.324610 | Durbin–Watson | 1.350773 | |

OLS, using observations 2000-01-06–2020-04-30 (T = 5179) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −0.000255606 | 7.97034 × 10${}^{-6}$ | −32.07 | 0.0000 |

STOXXSV_1 | 0.000249773 | 4.26226 × 10${}^{-6}$ | 58.60 | 0.0000 |

Mean dependent var | 0.000161 | S.D. dependent var | 0.000336 | |

Sum squared resid | 0.000351 | S.E. of regression | 0.000260 | |

${R}^{2}$ | 0.398798 | Adjusted ${R}^{2}$ | 0.398682 | |

$F(1,5177)$ | 3434.088 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.334086 | Durbin–Watson | 1.331768 | |

OLS, using observations 2000-01-05–2020-04-30 (T = 5179) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | −3.19171 × 10${}^{-6}$ | 4.70679 × 10${}^{-6}$ | −0.6781 | 0.4977 |

h2STOXX50_1 | 0.783757 | 0.0140212 | 55.90 | 0.0000 |

Mean dependent var | 0.000161 | S.D. dependent var | 0.000336 | |

Sum squared resid | 0.000364 | S.E. of regression | 0.000265 | |

${R}^{2}$ | 0.376384 | Adjusted ${R}^{2}$ | 0.376264 | |

$F(1,5177)$ | 3124.586 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.342843 | Durbin–Watson | 1.314043 | |

OLS, using observations 2000-02-02–2020-04-30 (T = 5160) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | ct-ratio | p-value | |

const | 2.05927 × 10${}^{-5}$ | 4.29557 × 10${}^{-6}$ | 4.794 | 0.0000 |

sq_DMSTOXXRET_1 | 0.118968 | 0.00666728 | 17.84 | 0.0000 |

sq_DMSTOXXRET_2 | 0.120534 | 0.00666684 | 18.08 | 0.0000 |

sq_DMSTOXXRET_3 | 0.0933684 | 0.00669582 | 13.94 | 0.0000 |

sq_DMSTOXXRET_4 | 0.0896290 | 0.00678649 | 13.21 | 0.0000 |

sq_DMSTOXXRET_5 | 0.0342222 | 0.00682410 | 5.015 | 0.0000 |

sq_DMSTOXXRET_6 | 0.0105941 | 0.00684945 | 1.547 | 0.1220 |

sq_DMSTOXXRET_7 | 0.0193777 | 0.00684880 | 2.829 | 0.0047 |

sq_DMSTOXXRET_8 | 0.0415985 | 0.00685618 | 6.067 | 0.0000 |

sq_DMSTOXXRET_9 | 0.0676568 | 0.00686950 | 9.849 | 0.0000 |

sq_DMSTOXXRET_10 | 0.0648663 | 0.00687561 | 9.434 | 0.0000 |

sq_DMSTOXXRET_11 | 0.00745897 | 0.00687600 | 1.085 | 0.2781 |

sq_DMSTOXXRET_12 | 0.00363238 | 0.00687086 | 0.5287 | 0.5971 |

sq_DMSTOXXRET_13 | 0.00415015 | 0.00685846 | 0.6051 | 0.5451 |

sq_DMSTOXXRET_14 | 0.00309319 | 0.00685311 | 0.4514 | 0.6518 |

sq_DMSTOXXRET_15 | 0.0459385 | 0.00685415 | 6.702 | 0.0000 |

sq_DMSTOXXRET_16 | −0.0110292 | 0.00682975 | −1.615 | 0.1064 |

sq_DMSTOXXRET_17 | 3.58184 × 10${}^{-5}$ | 0.00680192 | 0.005266 | 0.9958 |

sq_DMSTOXXRET_18 | −0.0226702 | 0.00671209 | −3.378 | 0.0007 |

sq_DMSTOXXRET_19 | 0.00604883 | 0.00668237 | 0.9052 | 0.3654 |

sq_DMSTOXXRET_20 | −0.0196779 | 0.00668193 | −2.945 | 0.0032 |

Mean dependent var | 0.000161 | S.D. dependent var | 0.000336 | |

Sum squared resid | 0.000324 | S.E. of regression | 0.000251 | |

${R}^{2}$ | 0.445202 | Adjusted ${R}^{2}$ | 0.443043 | |

$F(20,5139)$ | 206.1915 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.216381 | Durbin–Watson | 1.567234 |

Method | Test Statistic | Probability |
---|---|---|

S&P500 | ||

STOCHVOL | 1.06838 × 10${}^{11}$ | 0.0 |

GARCH | 1.01522 × 10${}^{11}$ | 0.0 |

sq_DMSPRET-1 | 2205.89 | 0.0 |

DOWJONES | ||

STOCHVOL | 7.72384 × 10${}^{10}$ | 0.0 |

GARCH | 7.33155 × 10${}^{10}$ | 0.0 |

sq_DMDJRET-1 | 1.84768 × 10${}^{16}$ | 0.0 |

STOXX50 | ||

STOCHVOL | 626.635 | 0.0 |

GARCH | 2.7139 × 10${}^{7}$ | 0.0 |

sq_DMSTOXRET-1 | 5449.58 | 0.0 |

Summary of 1000 MCMC draws after burn in of 1000 | |||||
---|---|---|---|---|---|

Prior Distributions | |||||

$\mu \sim normal$ | mean = 0 | S.D. = 100 | |||

$(\varphi +1)/2\sim \beta $ | ${a}_{0}=5$ | ${b}_{0}=1.5$ | |||

${\sigma}^{2}\sim {\chi}^{2}(df=1)$ | |||||

S&P500 | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −0.78 | 0.1562 | −1.05 | −0.78 | −0.53 |

$\varphi $ | 0.97 | 0.0048 | 0.96 | 0.97 | 0.97 |

$\sigma $ | 0.36 | 0.0182 | 0.32 | 0.36 | 0.39 |

$exp(\mu /2)$ | 0.68 | 0.0529 | 0.59 | 0.68 | 0.77 |

${\sigma}^{2}$ | 0.13 | 0.0130 | 0.11 | 0.13 | 0.15 |

DOWJONES | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −0.907 | 0.1447 | −1.14 | −0.913 | −0.656 |

$\varphi $ | 0.957 | 0.0058 | 0.95 | 0.957 | 0.966 |

$\sigma $ | 0.437 | 0.0232 | 0.40 | 0.435 | 0.478 |

$exp(\mu /2)$ | 0.637 | 0.0462 | 0.56 | 0.634 | 0.720 |

${\sigma}^{2}$ | 0.191 | 0.0204 | 0.16 | 0.189 | 0.229 |

STOXX50 | |||||

Posterior draws thinning = 1 | |||||

Mean | S.D. | 5% | 50% | 95% | |

$\mu $ | −0.37 | 0.1197 | −0.57 | −0.37 | −0.18 |

$\varphi $ | 0.95 | 0.0062 | 0.94 | 0.95 | 0.96 |

$\sigma $ | 0.41 | 0.0211 | 0.37 | 0.41 | 0.44 |

$exp(\mu /2)$ | 0.83 | 0.0494 | 0.75 | 0.83 | 0.92 |

${\sigma}^{2}$ | 0.17 | 0.0173 | 0.14 | 0.17 | 0.20 |

**Table 7.**Regression analysis of the three augmented stochastic volatility and GARCH models as explanators of RV5.

S&P500 | ||||
---|---|---|---|---|

OLS, using observations 2000-01-05–2020-04-30 (T = 5098) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 4.80570 × 10${}^{-5}$ | 2.59906 × 10${}^{-6}$ | 18.49 | 0.0000 |

SVREGSPRET_1 | 5.61407 × 10${}^{-7}$ | 6.87701 × 10${}^{-9}$ | 81.64 | 0.0000 |

Mean dependent var | 0.000112 | S.D. dependent var | 0.000269 | |

Sum squared resid | 0.000160 | S.E. of regression | 0.000177 | |

${R}^{2}$ | 0.566679 | Adjusted ${R}^{2}$ | 0.566594 | |

$F(1,5096)$ | 6664.347 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.288688 | Durbin–Watson | 1.422414 | |

OLS, using observations 2000-01-05–2020-04-30 (T = 5098) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.000136717 | 3.02817 × 10${}^{-6}$ | 45.15 | 0.0000 |

yhat3_1 | −0.0810177 | 0.00148855 | −54.43 | 0.0000 |

Mean dependent var | 0.000112 | S.D. dependent var | 0.000269 | |

Sum squared resid | 0.000233 | S.E. of regression | 0.000214 | |

${R}^{2}$ | 0.367610 | Adjusted ${R}^{2}$ | 0.367486 | |

$F(1,5096)$ | 2962.324 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.248686 | Durbin–Watson | 1.502451 | |

DOWJONES | ||||

OLS, using observations 2000-01-05–2020-04-30 (T = 5094) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 7.08759 × 10${}^{-5}$ | 3.11272 × 10${}^{-6}$ | 22.77 | 0.0000 |

DJSVREG_1 | 3.42022 × 10${}^{-7}$ | 5.52569 × 10${}^{-9}$ | 61.90 | 0.0000 |

Mean dependent var | 0.000114 | S.D. dependent var | 0.000287 | |

Sum squared resid | 0.000239 | S.E. of regression | 0.000217 | |

${R}^{2}$ | 0.429352 | Adjusted ${R}^{2}$ | 0.429240 | |

$F(1,5092)$ | 3831.187 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.382376 | Durbin–Watson | 1.235179 | |

OLS, using observations 2000-01-06–2020-04-30 (T = 5093) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.000144841 | 2.96368 × 10${}^{-6}$ | 48.87 | 0.0000 |

yhat7_1 | −0.00213366 | 3.18142 × 10${}^{-5}$ | −67.07 | 0.0000 |

Mean dependent var | 0.000114 | S.D. dependent var | 0.000287 | |

Sum squared resid | 0.000222 | S.E. of regression | 0.000209 | |

${R}^{2}$ | 0.469072 | Adjusted ${R}^{2}$ | 0.468968 | |

$F(1,5091)$ | 4497.872 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.347045 | Durbin–Watson | 1.305745 | |

OLS, using observations 2000-01-05–2020-04-30 (T = 5179) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 3.92593 × 10${}^{-5}$ | 3.84434 × 10${}^{-6}$ | 10.21 | 0.0000 |

STOXXREGFAC_1 | 1.04919 × 10${}^{-6}$ | 1.55219 × 10${}^{-8}$ | 67.59 | 0.0000 |

Mean dependent var | 0.000161 | S.D. dependent var | 0.000336 | |

Sum squared resid | 0.000310 | S.E. of regression | 0.000245 | |

${R}^{2}$ | 0.468805 | Adjusted ${R}^{2}$ | 0.468702 | |

$F(1,5177)$ | 4568.947 | p-value(F) | 0.000000 | |

$\widehat{\rho}$ | 0.212647 | Durbin–Watson | 1.574424 | |

OLS, using observations 2000-01-05–2020-04-30 (T = 5179) | ||||

Dependent variable: rv5 | ||||

Coefficient | Std. Error | t-ratio | p-value | |

const | 0.000191985 | 4.32834 × 10${}^{-6}$ | 44.36 | 0.0000 |

yhat2_1 | −0.111512 | 0.00332039 | −33.58 | 0.0000 |

Mean dependent var | 0.000161 | S.D. dependent var | 0.000336 | |

Sum squared resid | 0.000479 | S.E. of regression | 0.000304 | |

${R}^{2}$ | 0.178889 | Adjusted ${R}^{2}$ | 0.178731 | |

$F(1,5177)$ | 1127.876 | p-value(F) | 7.0 × 10${}^{-224}$ | |

$\widehat{\rho}$ | 0.258795 | Durbin–Watson | 1.482403 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Allen, D.E.
Stochastic Volatility and GARCH: Do Squared End-of-Day Returns Provide Similar Information? *J. Risk Financial Manag.* **2020**, *13*, 202.
https://doi.org/10.3390/jrfm13090202

**AMA Style**

Allen DE.
Stochastic Volatility and GARCH: Do Squared End-of-Day Returns Provide Similar Information? *Journal of Risk and Financial Management*. 2020; 13(9):202.
https://doi.org/10.3390/jrfm13090202

**Chicago/Turabian Style**

Allen, David Edmund.
2020. "Stochastic Volatility and GARCH: Do Squared End-of-Day Returns Provide Similar Information?" *Journal of Risk and Financial Management* 13, no. 9: 202.
https://doi.org/10.3390/jrfm13090202